What does "continuous spacetime" mean?

Question

I often encounter discussions, such as seen here, about whether spacetime is discrete or continuous. However, I am only familiar with continuity as being a property of functions. I saw this question about continuous spaces, but I didn't find a definitive answer there, so I ask what is meant when we say that spacetime is "continuous?" Does it mean that its locally homeomorphic to $\mathbb R^{3+1}$ or something else?

Answer 1

You are right. Spacetime is defined as a Manifold

A manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an $n$-dimensional manifold, or $n$-manifold for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to the Euclidean space of dimension $n$.

More precisely, spacetime is a Lorentzian manifold, for which replace Euclidean space with Minkowski spacetime.